No, that privilege should be just for the child.

Parents inhabit a closed world with their children, and there is a lot of abuse that takes place with no one looking on. That used to be completely private. However, now laws are favoring the child being able to tell what is going on, which is as it should be. The parent should have no role in this privilege.

Parent's do not have a right to their children.

There is not currently a parent and child privilege in the United States. There shouldn't be one either. Some parents are simply not good for their children. Parents who have harmed their children either physically or mentally have no right to their children. Neither do parents who have given their children up for adoption. The important focus is the well-being of the child. It should never what is convenient to the parent.